The Transient Universe is abundant with sources that serve as exceptional laboratories for studying radiation processes in multiple windows, as evidenced by UHECRs, black holes in AGN and microquasars, SN1987A, Cygnus X-1, PSR 1913+16, CC-SNe and GRBs. As general relativity is becoming a genuine experimental science with the LAGEOS and Gravity Probe B experiments beyond mere redshift effects, and gravitational-wave and neutrino experiments are advancing to next generation sensitivity, this decade is expected to bring major new discoveries with an inevitable transformation of our understanding of their astronomical origin and the physics of their radiation processes.

This pursuit requires a concerted effort on effective observational strategies, theory, advanced data analysis and high-performance computing, including integration of the three different areas of electromagnetic, hadronic and gravitational radiation processes. We hope that the present book offers a useful introduction to this exciting development, for those who wish to pick up this challenge.

In the electromagnetic spectrum, radiation processes tend to extend far out, while neutrino and gravitational-wave emissions often tend to be confined to or nearby the energy source driving these emissions. The relation between these various windows of observations thereby tends to be non-trivial, also because the most relativistic sources are transient. It generally calls for time-dependent models that elucidate correlations in energies, time scales, light curves, the associated scaling relations and, possibly, normalizations.

UHECRs and GRBs discussed in the previous chapters are some of the most mysterious discoveries of the last century. Their astronomical origin has only recently been constrained by the PAO and various satellite missions since the discovery of X-ray afterglows by Beppo-SAX.

Black holes are natural candidates for powering these emissions. Frame dragging around rotating black holes acts universally on particles and fields alike, which opens a broad range of channels in non-thermal emissions. Furthermore, black holes are scale free, with no intrinsic reference to a particular mass, in sharp contrast to degenerate compact objects, i.e., neutron stars and white dwarfs (see Chapter 1).

Extracting evidence for black holes as inner engines powering these emissions requires detailed analysis of all their radiation channels, taking into account a large diversity in phenomenology as expressed by supermassive and stellar mass black holes in view of their scale-free behavior. Scale-free behavior may also be expressed in ensembles of specific types of sources, provided the ensembles are sufficiently large in number.

Alfvén waves in transient capillary jets

When the magnetosphere around rotating black holes is intermittent, e.g., due to instabilities in the disk or the inner torus magnetosphere [599], magnetic outflows produce terminal Alfvén fronts propagating along their spin axis out to large distances. The approximation of ideal MHD discussed in Chapter 6 assumes negligible dissipation of the electromagnetic field in the fluid, corresponding to an infinite magnetic Reynolds number, which applies to extragalactic radio jets [203].

The aim of this book is to bridge the considerable gap that exists between standard undergraduate mechanics texts, which rarely cover topics in celestial mechanics more advanced than two-body orbit theory, and graduate-level celestial mechanics texts, such as the well-known books by Moulton (1914), Brouwer and Clemence (1961), Danby (1992), Murray and Dermott (1999), and Roy (2005). The material presented here is intended to be intelligible to an advanced undergraduate or beginning graduate student with a firm grasp of multivariate integral and differential calculus, linear algebra, vector algebra, and vector calculus.

The book starts with a discussion of the fundamental concepts of Newtonian mechanics, as these are also the fundamental concepts of celestial mechanics. A number of more advanced topics in Newtonian mechanics that are needed to investigate the motions of celestial bodies (e.g., gravitational potential theory, motion in rotating reference frames, Lagrangian mechanics, Eulerian rigid body rotation theory) are also described in detail in the text. However, any discussion of the application of Hamiltonian mechanics, Hamilton-Jacobi theory, canonical variables, and action-angle variables to problems in celestial mechanics is left to more advanced texts (see, for instance, Goldstein, Poole, and Safko 2001).

Celestial mechanics (a term coined by Laplace in 1799) is the branch of astronomy that is concerned with the motions of celestial objects—in particular, the objects that make up the solar system—under the influence of gravity.

Introduction

This chapter describes an elegant reformulation of the laws of Newtonian mechanics that is due to the French-Italian scientist Joseph Louis Lagrange (1736–1813). This reformulation is particularly useful for finding the equations of motion of complicated dynamical systems.

Generalized coordinates

Let the qi, for i = 1, ℱ, be a set of coordinates that uniquely specifies the instantaneous configuration of some dynamical system. Here, it is assumed that each of the qi can vary independently. The qi might be Cartesian coordinates, angles, or some mixture of both types of coordinate, and are therefore termed generalized coordinates. A dynamical system whose instantaneous configuration is fully specified by ℱ independent generalized coordinates is said to have ℱ degrees of freedom. For instance, the instantaneous position of a particle moving freely in three dimensions is completely specified by its three Cartesian coordinates, x, y, and z. Moreover, these coordinates are clearly independent of one another. Hence, a dynamical system consisting of a single particle moving freely in three dimensions has three degrees of freedom. If there are two freely moving particles then the system has six degrees of freedom, and so on.

Suppose that we have a dynamical system consisting of N particles moving freely in three dimensions. This is an ℱ = 3 N degree-of-freedom system whose instantaneous configuration can be specified by F Cartesian coordinates. Let us denote these coordinates the xj, for j = 1, ℱ.

Introduction

The orbital motion of the planets around the Sun is fairly accurately described by Kepler's laws. (See Chapter 3.) Similarly, to a first approximation, the orbital motion of the Moon around the Earth can also be accounted for via these laws. However, unlike the planetary orbits, the deviations of the lunar orbit from a Keplerian ellipse are sufficiently large that they are easily apparent to the naked eye. Indeed, the largest of these deviations, which is generally known as evection, was discovered in ancient times by the Alexandrian astronomer Claudius Ptolemy (90 BCE–168 CE) (Pannekoek 2011). Moreover, the next largest deviation, which is called variation, was first observed by Tycho Brahe (1546–1601) without the aid of a telescope (Godfray 1853). Another non-Keplerian feature of the lunar orbit, which is sufficiently obvious that it was known to the ancient Greeks, is the fact that the lunar perigee (i.e., the point of closest approach to the Earth) precesses (i.e., orbits about the Earth in the same direction as the Moon) at such a rate that, on average, it completes a full circuit every 8.85 years. The ancient Greeks also noticed that the lunar ascending node (i.e., the point at which the Moon passes through the fixed plane of the Earth's orbit around the Sun from south to north) regresses (i.e., orbits about the Earth in the opposite direction to the Moon) at such a rate that, on average, it completes a full circuit every 18.6 years (Pannekoek 2011).

Introduction

Newtonian mechanics is a mathematical model whose purpose is to account for the motions of the various objects in the universe. The general principles of this model were first enunciated by Sir Isaac Newton in a work titled Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). This work, which was published in 1687, is nowadays more commonly referred to as the Principia.

Until the beginning of the twentieth century, Newtonian mechanics was thought to constitute a complete description of all types of motion occurring in the universe. We now know that this is not the case. The modern view is that Newton's model is only an approximation that is valid under certain circumstances. The model breaks down when the velocities of the objects under investigation approach the speed of light in a vacuum, and must be modified in accordance with Einstein's special theory of relativity. The model also fails in regions of space that are sufficiently curved that the propositions of Euclidean geometry do not hold to a good approximation, and must be augmented by Einstein's general theory of relativity. Finally, the model breaks down on atomic and subatomic length scales, and must be replaced by quantum mechanics. In this book, we shall (almost entirely) neglect relativistic and quantum effects. It follows that we must restrict our investigations to the motions of large (compared with an atom), slow (compared with the speed of light) objects moving in Euclidean space. Fortunately, virtually all the motions encountered in conventional celestial mechanics fall into this category.

Introduction

We saw earlier, in Section 1.9, that an isolated dynamical system consisting of two freely moving point masses exerting forces on one another—which is usually referred to as a two-body problem—can always be converted into an equivalent one-body problem. In particular, this implies that we can exactly solve a dynamical system containing two gravitationally interacting point masses, as the equivalent one-body problem is exactly soluble. (See Sections 1.9 and 3.16.) What about a system containing three gravitationally interacting point masses? Despite hundreds of years of research, no useful general solution of this famous problem—which is usually called the three-body problem—has ever been found. It is, however, possible to make some progress by severely restricting the problem's scope.

Circular restricted three-body problem

Consider an isolated dynamical system consisting of three gravitationally interacting point masses, m1, m2, and m3. Suppose, however, that the third mass, m3, is so much smaller than the other two that it has a negligible effect on their motion. Suppose, further, that the first two masses, m1 and m2, execute circular orbits about their common center of mass. In the following, we shall examine this simplified problem, which is usually referred to as the circular restricted three-body problem. The problem under investigation has obvious applications to the solar system. For instance, the first two masses might represent the Sun and a planet (recall that a given planet and the Sun do indeed execute almost circular orbits about their common center of mass), whereas the third mass might represent an asteroid or a comet (asteroids and comets do indeed have much smaller masses than the Sun or any of the planets).

Introduction

The two-body orbit theory described in Chapter 3 neglects the direct gravitational interactions between the planets, while retaining those between each individual planet and the Sun. This is an excellent first approximation, as the former interactions are much weaker than the latter, as a consequence of the small masses of the planets relative to the Sun. (See Table 3.1.) Nevertheless, interplanetary gravitational interactions do have a profound influence on planetary orbits when integrated over long periods of time. In this chapter, a branch of celestial mechanics known as orbital perturbation theory is used to examine the secular (i.e., long-term) influence of interplanetary gravitational perturbations on planetary orbits. Orbital perturbation theory is also used to investigate the secular influence of planetary perturbations on the orbits of asteroids, as well as the secular influence of the Earth's oblateness on the orbits of artificial satellites.

Evolution equations for a two-planet solar system

For the moment, let us consider a simplified solar system that consists of the Sun and two planets. (See Figure 9.1.) Let the Sun be of mass M and position vector Rs. Likewise, let the two planets have masses m and m′ and position vectors R and R′, respectively. Here, we are assuming that m, m′ ≪ M. Finally, let r = R − Rs and r′ = R′ − Rs be the position vectors of each planet relative to the Sun. Without loss of generality, we can assume that r′ > r.